According to the OED, there is an ancient connection between English
'let' "allow" and 'let' "hinder", = ON 'láta' "let, cause to" and
'letja' "hold someone back, [try to] dissuade". Also between these
and the English adjective 'late', and its ON cognate 'latr' "slow,
lazy". The difference in vowels would be due to ablaut (vowel
gradation), i.e. caused by stress differences in Proto-Indo-European,
and (in the case of 'letja') by i-umlaut = i-mutation, compare Gothic
'latjan' "to delay, to hold up", and the Old Norse past tense of
'letja', 'latti' which preserves the unmutated vowel.

The thing I was curious about though was the position of the pronoun
in 'mun mig eigi tjóa að letja', since this is apparently the
complement (direct object) of the letja-clause. By the way, this gets
a bit involved, and isn't really necessary for understanding the
sentence or reading Njáls saga, so don't get bogged down in these
niggling details if they're not to your taste!

Anyway, I've just had a chance to check in Faarlund's "The Syntax of
Old Norse", and found the following:

"When there are two non-finite verbs in the sentence the normal order
is auxiliary verb-main verb in accordance with the VO (Verb Object)
pattern (1); it is not uncommon however for the non-finite auxiliary
to follow the main verb (2). [...] If the main verb in such a
construction has a complement, this complement may precede its head,
yielding a consistent OV word order (3). [...] The complement may
also follow its auxiliary (4), but it never follows the main verb
directly (5). [...] The reason this is disallowed may be that the
main verb plus the auxiliary was reanalysed as one verbal head after
the OV (Object Verb) order was no longer productive. The
main-auxiliary collocation consequently behaves as one syntactic word,
which allows nothing to intervene." (Faarlund: The Syntax of Old
Norse, 162-163, 8.4.6 "Head-complement order").

1. mun eigi vilja af þér taka
2. því er þú vilt spurt hafa
3. þú vilt honum þjónat hafa
4. hann mun ráða vilja ferðum sínum
5. * hann mun ráða ferðum sínum vilja

(The asterisk * is used by syntactitians to indicate a wrongly
constructed sentence, a usage which doesn't occur in the language
under discussion.)

But our example doesn't quite quite fit any of these patterns, since
(a) 'tjóa' isn't an auxiliary verb, (b) the non-finite verbs are
arranged in VO order, and yet (c) the complement of 'letja' is still
placed before both non-finite verbs.