'Luke' in the compound 'lukewarm' functions as a downtoning semantic reduplication, supposing 'luke' is considered to have the (perhaps) more modern sense of "a moderate degree of warmth, tepid" rather than the (perhaps) earlier sense of "weak, little". In some other compounds containing 'luke', 'luke' downtones the second element. Examples of such downtoning, although thin on the ground, include 'luke-hearted' (and, possibly, 'Lukeheart') and 'luke hot'.
In English etymology, semantic reduplication is mentioned by, for example, Liberman (An Analytic Dictionary of English Etymology: An Introduction, 2007), with reference to 'ragamuffin' (emphasis mine):
The most convincing hypothesis traces both rag- and -muffin to words for 'devil,' as in OF Rogomant (though in French it may have been a borrowing from Germanic), preserved in E Ragman and Ragman's roll (> rigmarole), and Old Muffy, from AF maufé 'ugly; the Evil One.' Ragamuffin is then a semantic reduplication with an augment (-a-) in the middle, *'devil-a-devil.'
A 'downtoner' and so 'to downtone', in the linguistic sense, approximate the opposites of 'intensifier' and 'to intensify'. Both intensifiers and downtoners are qualifiers. Intensifiers used ironically sometimes function as downtoners; often, negating an intensifier converts it into a downtoner.
Hence, 'luke' in the sense of "tepid, moderately warm" downtones 'warm' in the compound 'lukewarm'.
For the etymological influence on 'luke' of Anglo-Saxon wlæc, the older meaning of which was 'weak', see Skeat 1882, An etymological dictionary of the English language (under the headword 'lukewarm', after the pilcrow). This influence suggests the sense of 'lukewarm' may derive from elements meaning "weak-warm", a meaning which accurately represents contemporary use.
For a suggestion that 'luke' was used dialectally in the sense of 'little' see Cockeram 1623, An English dictionarie, as cited by Robertson in Life on the upper Thames (middle of last paragraph, p 19, where "Cockerham" is either a spelling variant or an error). Dialectal use of 'luke' with the meaning "little, slight", when compounded with 'warm' into 'lukewarm', contributed to the contemporary sense of 'lukewarm': "slightly warm, a little warm".
As mentioned and exampled by Skeat 1882 (op. cit.) from Dickens, etc.,
Luke means 'tepid' and can correctly be used alone,
as well as to downtone or otherwise qualify various signifiers: warm, blood, bath, water, fire, hot, and heart.
Some Paywalled Attestations from the 20th Century
'Luke-hearted' survives primarily as a family or nickname. One example of the latter is this "Personals" ad from The Los Angeles Times, 18 May 1960:
PATRICIA Gentile, aka Lukeheart, call MA. 5-1494. Urgent.
Attestation of 'Lukeheart' as a family name is too common to be worth exampling. Other attestations of 'luke-hearted' and variants are vanishingly infrequent in the 20th century. They include a coinage in The Times Herald (Port Huron, Michigan) 31 Jul 1967:
Naturally, the two find ways to help each other in this luke-heartwarming episode [of the television show "Coronet Blue"].
Another instance is this from The Courier-Journal (Louisville, Kentucky)
20 Jul 1988, reprinted a week later in two other newspapers, then 9 years (1997) later in the original venue:
People used to refer to nice folks as luke-hearted, which means, of course, warm-hearted. But that word seems to have disappeared, which is too bad.
The author of that article seems to have misconstrued the usual sense of 'luke-hearted' in the 19th century, which was more akin to a much-weakened or neutral 'disheartened' than "warm-hearted".
Examples of 'luke-hot' from the 20th century are more various, as follows.
In the Daily News (New York, New York), 17 Feb 1946, the intended meaning of 'luke-hot' may diverge from the expectable sense of "slightly or weakly hot":
June introduced some luke-hot letters which indicated that Schreiner's ardor had been considerable.
The meaning is as expected in the Edmonton Journal (Edmonton, Alberta, Canada) 01 Apr 1964, but I hesitate to ascribe that accuracy to a hypothesized more-literate Canadian audience:
Even if you have failed to see the Oilers play this season, you know they've been white-hot because the cliche-riddled news services have been calling them the white-hot Oil Kings since last October.
When will the news people stop? As soon as the Kings demote themselves to grey-hot, or luke-hot.
From The Miami News (Miami, Florida) 15 May 1968:
In fact, my independent survey showed these percentages of old-school, step-outside-and-say-that, rabid patisans in either party:
McCarthy, two per cent. Kennedy, two per cent. Nixon, two per cent. Rockefeller, two per cent. Utterly baffled, 91 per cent. (The other one per cent is still for Goldwater.)
This doesn't include the great numbers of luke-hot supporters, of course: The millions who stand around in the booth and eventually vote for somebody....
From The News Journal (Wilmington, Delaware) 23 Jan 1973:
Delaware's last governor got into political hot water. If it happens again, the experience should be less painful, for the state is going to reduce the temperature of the water that comes out of its "hot" faucets.
Would you believe a future administration getting into luke-hot water?
From The Sydney Morning Herald (Sydney, New South Wales, Australia) 12 Jun 1989 (reprinted with slight editing in 1993):
...James Woods, as the shady nightclub owner Jake Wise [in the movie Against All Odds], delivers a nice performance. But apart from a few luke-hot scenes between Ward and Bridges that gave the flick its R rating on the big screen....
From the Santa Maria Times (Santa Maria, California) 21 Jan 1991 (the "Luke-hot List" trope was repeated for several weeks in the column):
To add to this tangle, production controllers also invented six manual lists to override the formal systems: the Hot List (angry customers), the Luke-hot List (moderately angry customers), ....
In the 'luke-water' compound, 'luke' is used with the sense of "slightly or weakly warm". Following are six examples of the compound from the 20th and 21st century popular press.
Belle Plaine News (Belle Plaine, Kansas) 03 Nov 1910:
...wading and bathing in the luke water of the Gulf of Mexico.
Akron Evening Times (Akron, Ohio)
15 Apr 1920:
For every two gallons of warm (LUKE) water you may need to add one-half pint of the above mixture, ....
The Journal News (White Plains, New York)
03 Mar 1952:
...add a pinch of soda dissolved in luke water water....
McKinney Weekly Democrat-Gazette (Mckinney, Texas)
05 Jul 1956:
A good way to remove chocolate stain is to pour luke water from a height of about two feet through the stained area of material.
Janesville Daily Gazette (Janesville, Wisconsin)
09 Nov 1967:
The easiest way to remove the skunk odor is to wash the dog thoroughly in luke water with plenty of Lifebuoy soap.
Calgary Herald (Calgary, Alberta, Canada)
02 Dec 2001 (headline):
Mark Wahlberg in luke water.
Attestation of the compounds 'luke-hearted', 'luke hot', 'luke water', and 'luke bath' appears in the 19th and earlier centuries. For an example of the last, this from the 1880 Textile colorist:
...with methyl-green finish dyeing in a luke bath.
Attestation of 'luke' qualifying 'blood', and 'fire', absent compounding, appear in the OED citation collection. They can be seen without going through a paywall in the earlier NED, under the headword 'luke'.