A file can be downloaded from the University of Sussex pertaining to this: Box in English place-names.
Here's an excerpt to tempt you:
"A Roman villa site hasbeen identified at Pincent's Farm in Tilehurst (Scott BK 42), and it might also be considered interesting that box-fronds are said to have been recovered from a Roman context at Silchester, some six miles away, as mentioned in the Introduction (0.1). 2.4.11 Boxholt (Kirdford, Sussex; lost, site unknown; PN Sussex 103) For the element holt, see above under Boxholt in St John Without (Lewes)(2.1.8). Boxalland Farm is just north of Kirdford village, but it is far from certain that this is the site of Boxholt. It takes its name from the family of William Boxall, who had woods of oak and beech locally in 1656 (Kenyon 1955: 141). The surname presumably derives from the lost Boxholt (cf.McKinley 1988: 119 and 137, footnote 164), but that is no guarantee that Boxalland is Boxholt. The same family probably spread and gave its name to places in nearby Lurgashall and Linchmere. Since the place is lost, nothing can be said about its geology beyond the fact that the parish is Wealden and that the strata outcropping over most of the parish are the Weald Clay. David Streeter comments (personal communication) that the area is on a slight risekeeping it above the Wealden frost-hollows, which is specially relevant if Boxholt was west of Kirdford; Kirdford village itself is situated low. 2.4.12 Boxlees Hill (Minster-in-Thanet, Kent; TR 306631) This is recorded (Cullen 1996) as Eastboxlemershe (c14), Boxile (before 1410-4). It may derive its name directly or indirectly from that of Boxley (1.3) despite the apparent early lack of genitival construction. Perhaps some day a connection with Boxley Abbey may be shown; but since this has not yet occurred, it is safest to regard the name as independent. This once-coastal marshland site can, on classical assumptions, never have been native box country, for the ecological reasons spelt out above (Section 1). But there is a probable villa site in the parish (Scott KE 67, at TR 3164)."
(Please note, I have a copy of this document archived, and available on request, in case the site ever goes down, because I feel it is tremendously important).
Early versions of the name of Boxall vary considerably, so don't be alarmed by spellings in the list below, especially in medieval times. My children find the version Boxhole very appealing, it reminds them of "Hobbit Hole", and a sign thus has gone up over their "camp" (a Box-elder tree, chosen by sheer coincidence, but now known as the "Boxallder tree").